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grade7

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(2)

grade 7 problems (IV Soros Olympiad 1997-98 Correspondence Round)

Source:

5/31/2024
p1. The oil pipeline passes by three villages AA, BB, CC. In the first village, 30%30\% of the initial amount of oil is drained, in the second - 40%40\% of the amount that will reach village BB, and in the third - 50%50\% of the amount that will reach village CC What percentage of the initial amount of oil reaches the end of the pipeline?
p2. There are several ordinary irreducible fractions (not necessarily proper) with natural numerators and denominators (and the denominators are greater than 11). The product of all fractions is equal to 1010. All numerators and denominators are increased by 11. Can the product of the resulting fractions be greater than 1010?
p3. The garland consists of 1010 light bulbs connected in series. Exactly one of the light bulbs has burned out, but it is not known which one. There is a suitable light bulb available to replace a burnt out one. To unscrew a light bulb, you need 1010 seconds, to screw it in - also 1010 seconds (the time for other actions can be neglected). Is it possible to be guaranteed to find a burnt out light bulb: a) in 1010 minutes, b) in 55 minutes?
p4. When fast and slow athletes run across the stadium in one direction, the fast one overtakes the slow one every 1515 minutes, and when they run towards each other, they meet once every 55 minutes. How many times is the speed of a fast runner greater than the speed of a slow runner?
p5. Petya was 3535 minutes late for school. Then he decided to run to the kiosk for ice cream. But when he returned, the second lesson had already begun. He immediately ran for ice cream a second time and was gone for the same amount of time. When he returned, it turned out that he was late again, and he had to wait 5050 minutes before the start of the fourth lesson. How long does it take to run from school to the ice cream stand and back if each lesson, including recess after it, lasts 5555 minutes?
p6. In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon.
p7. In the writing of the antipodes, numbers are also written with the digits 0,...,90, ..., 9, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes 5∗8+7+1=485 * 8 + 7 + 1 = 48 2∗2∗6=242 * 2 * 6 = 24 5∗6=305* 6 = 30 a) How will the equality 23=...2^3 = ... in the writing of the antipodes be continued? b) What does the number 9 mean among the Antipodes?
Clarifications: a) It asks to convert 232^3 in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit 99 mean among the antipodes, i.e. with which digit is it equal in our number system?
p8. They wrote the numbers 1,2,3,4,...,1996,19971, 2, 3, 4, ..., 1996, 1997 in a row. Which digits were used more when writing these numbers - ones or twos? How long?
p9. On the number axis there lives a grasshopper who can jump 11 and 44 to the right and left. Can he get from point 11 to point 22 of the numerical axis in1996in 1996 jumps if he must not get to points with coordinates divisible by 4 4 (points 00, ±4\pm 4, ±8\pm 8, etc.)?
p10. Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
algebrageometrycombinatoricsnumber theorySoros Olympiad
grade 7 problems (IV Soros Olympiad 1997-98 Round 2)

Source:

5/31/2024
p1. In the correct identity (x2−1)(x+...)=(x+3)(x−1)(x+...)(x^2 - 1)(x + ...) = (x + 3)(x- 1)(x +...) two numbers were replaced with dots. What were these numbers?
p2. A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take 10%10\% of the amount currently available, on the other - 20%20\%, etc. . How should the merchant travel to bring as much of the money as possible to B? What part of the original amount will he bring to B? https://cdn.artofproblemsolving.com/attachments/f/5/ab62ce8fce3d482bc52b89463c953f4271b45e.png
p3. Find the angle between the hour and minute hands at 77 hours 3838 minutes.
p4. The lottery game is played as follows. A random number from 11 to 10001000 is selected. If it is divisible by 22, they pay a ruble, if it is divisible by 1010 - two rubles, by 1212 - four rubles, by 2020 - eight, if it is divisible by several of these numbers, then they pay the sum. How much can you win (at one time) in such a game? List all options.
p5.The sum of the digits of a positive integer xx is equal to nn. Prove that between xx and 10x10x you can find an integer whose sum of digits is n+5 n + 5.
p6. 99 people took part in the campaign, which lasted 1212 days. There were 33 people on duty every day. At the same time, the duty officers quarreled with each other and no two of them wanted to be on duty together ever again. Nevertheless, the participants of the campaign claim that for all 1212 days they were able to appoint three people on duty, taking into account this requirement. Could this be so?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
algebrageometrycombinatoricsnumber theorySoros Olympiad